Numerical solution of two-dimensional nonlinear Schrödinger equation using a new two-grid finite element method

Abstract

A new two-grid finite element scheme is presented for two-dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the nonlinear system using the coarse-grid solution as the initial guess, and furthermore one more linear system on the coarse space is solved. The error estimations of the two-grid solution in the L2 and H1 norm are given. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and two-grid algorithm still achieves optimal approximation as long as the mesh sizes satisfy H=O(h13).